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Why, you may ask?

The Python sympy symbolic library provides solutions to single, linear Diophantine equations in terms of parametric variables. For instance,

{(t_0, -9*t_0 - 5*t_1 + 154, -5*t_0 - 3*t_1 + 77)}

Without worrying about coding details, how can one solve such a system of parametric expressions for values of the original variables constrained to be, for instance, greater than zero.

Are there any opensource products?

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    $\begingroup$ Sage can do this most likely, although integer linear programming is generally NP-hard. $\endgroup$
    – Kirill
    Oct 26 '16 at 18:39
  • $\begingroup$ Thank you, yes, I understand. I'm asking because I would like to offer an example involving only a small number of variables, for sympy. $\endgroup$
    – Bill Bell
    Oct 26 '16 at 18:51
  • $\begingroup$ Can you explain what the syntax of the example means? To me, this simply reads like a collection of linear expressions. How does it represent a system of linear inequalities? $\endgroup$
    – Wolfgang Bangerth
    Oct 26 '16 at 19:51
  • $\begingroup$ @WolfgangBangerth: Say the Diophantine equation were given in terms of x, y and z. Then one could make arbitrary choice of integers t_0 and t_1 and then one tuple of values for x, y and z would be given by that tuple in the set. I would like to be able to solve, for instance, for the system of inequalities where each item in the tuple is set >0. $\endgroup$
    – Bill Bell
    Oct 26 '16 at 20:18
  • $\begingroup$ So is there only one solution, or is it a whole region in space that satisfies these equations? Are you only interested in one feasible solution, or in characterizing the whole region? $\endgroup$
    – Wolfgang Bangerth
    Oct 27 '16 at 14:02
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If you are looking for a numerical answer, LP and MILP solvers can efficiently (not always for MIP) provide you with a feasible solution by optimizing an objective function of zeroes.

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